Interpretable machine learning for high-dimensional trajectories of aging health.

We have built a computational model for individual aging trajectories of health and survival, which contains physical, functional, and biological variables, and is conditioned on demographic, lifestyle, and medical background information. We combine techniques of modern machine learning with an interpretable interaction network, where health variables are coupled by explicit pair-wise interactions within a stochastic dynamical system. Our dynamic joint interpretable network (DJIN) model is scalable to large longitudinal data sets, is predictive of individual high-dimensional health trajectories and survival from baseline health states, and infers an interpretable network of directed interactions between the health variables. The network identifies plausible physiological connections between health variables as well as clusters of strongly connected health variables. We use English Longitudinal Study of Aging (ELSA) data to train our model and show that it performs better than multiple dedicated linear models for health outcomes and survival. We compare our model with flexible lower-dimensional latent-space models to explore the dimensionality required to accurately model aging health outcomes. Our DJIN model can be used to generate synthetic individuals that age realistically, to impute missing data, and to simulate future aging outcomes given arbitrary initial health states.

Torsion and bistability of double-twist elastomers.

We investigate the elastic properties of anisotropic elastomers with a double-twist director field, which is a model for collagen fibrils or blue phases. We observe a significant Poynting-like effect, coupling torsion (fibril twist) and extension. For freely-rotating boundary conditions, we identify a structural bistability at very small extensional strains which undergoes a saddle-node bifurcation at a critical strain - at approximately 1% strain for a parameterization appropriate for collagen fibrils. With clamped boundary conditions appropriate for many experimental setups, the bifurcation is not present. We expect significant helical shape effects when fixed torsion does not equal the equilibrium torsion of freely-rotating boundary conditions, due to residual torques.

Non-equilibrium growth and twist of cross-linked collagen fibrils.

The lysyl oxidase (LOX) enzyme that catalyses cross-link formation during the assembly of collagen fibrils in vivo is too large to diffuse within assembled fibrils, and so is incompatible with a fully equilibrium mechanism for fibril formation. We propose that enzymatic cross-links are formed at the fibril surface during the growth of collagen fibrils; as a consequence no significant reorientation of previously cross-linked collagen molecules occurs inside collagen fibrils during fibril growth in vivo. By imposing local equilibrium only at the fibril surface, we develop a coarse-grained quantitative model of in vivo fibril structure that incorporates a double-twist orientation of collagen molecules and a periodic D-band density modulation along the fibril axis. Radial growth is controlled by the concentration of available collagen molecules outside the fibril. In contrast with earlier equilibrium models of fibril structure, we find that all fibrils can exhibit a core-shell structure that is controlled only by the fibril radius. At small radii a core is developed with a linear double-twist structure as a function of radius. Within the core the double-twist structure is largely independent of the D-band. Within the shell at larger radii, the structure approaches a constant twist configuration that is strongly coupled with the D-band. We suggest a stable radius control mechanism that corneal fibrils can exploit near the edge of the linear core regime; while larger tendon fibrils use a cruder version of growth control that does not select a preferred radius.

Chiral phase-coexistence in compressed double-twist elastomers.

We adapt the theory of anisotropic rubber elasticity to model cross-linked double-twist liquid crystal cylinders such as exhibited in biological systems. In mechanical extension we recover strain-straightening, but with an exact expression in the small twist-angle limit. In compression, we observe coexistence between high and low twist phases. Coexistence begins at small compressive strains and is robustly observed for any anisotropic cross-links and for general double-twist functions - but disappears at large twist angles. Within the coexistence region, significant compression of double-twist cylinders is allowed at constant stress. Our results are qualitatively consistent with previous observations of swollen or compressed collagen fibrils, indicating that this phenomenon may be readily accessible experimentally.

Bayesian counting of photobleaching steps with physical priors.

Counting fluorescence photobleaching steps is commonly used to infer the number n0 of monomeric units of individual oligomeric protein complexes or misfolded protein aggregates. We present a principled Bayesian approach for counting that incorporates the statistics of photobleaching. Our physics-based prior leads to a simple and efficient numerical scheme for maximum a posteriori probability (MAP) estimates of the initial fluorophore number n^0. Our focus here is on using a calibration to precisely estimate n^0, though our approach can also be used to calibrate the photophysics. Imaging noise increases with n^0, while bias is often introduced by temporal averaging. We examine the effects of fluorophore number n^0 of the oligomer or aggregate, lifetime photon yield µeff of an individual fluorophore, and exposure time Δt of each image frame in a time-lapse experiment. We find that, in comparison with standard ratiometric approaches or with a "change-point" step-counting algorithm, our MAP approach is both more precise and less biased.

Polymorphism of stable collagen fibrils.

Collagen fibrils are versatile self-assembled structures that provide mechanical integrity within mammalian tissues. The radius of collagen fibrils vary widely depending on experimental conditions in vitro or anatomical location in vivo. Here we explore the variety of thermodynamically stable fibril configurations that are available. We use a liquid crystal model of radial collagen fibril structure with a double-twist director field. Using a numerical relaxation method we show that two dimensionless parameters, the ratio of saddle-splay to twist elastic constants k24/K22 and the ratio of surface tension to chiral strength [small gamma, Greek, tilde] ≡ γ/(K22q), largely specify both the scaled fibril radius and the associated surface twist of equilibrium fibrils. We find that collagen fibrils are the stable phase with respect to the cholesteric phase only when the reduced surface tension is small, [small gamma, Greek, tilde] ⪅ 0.2. Within this stable regime, collagen fibrils can access a wide range of radii and associated surface twists. Remarkably, we find a maximal equilibrium surface twist of 0.33 rad (19°). Our results are compatible with corneal collagen fibrils, and we show how the large surface twist can explain the narrow distribution of corneal fibril radii. Conversely, we show how small surface twist is required for the thermodynamic stability of tendon fibrils in the face of considerable polydispersity of radius.

Photobleaching of randomly rotating fluorescently decorated particles.

Randomly rotating particles that have been isotropically labeled with rigidly linked fluorophores will undergo non-isotropic (patchy) photobleaching under illumination due to the dipole coupling of fluorophores with light. For a rotational diffusion rate D of the particle and a photobleaching time scale τ of the fluorophores, the dynamics of this process are characterized by the dimensionless combination Dτ. We find significant interparticle fluctuations at intermediate Dτ. These fluctuations vanish at both large and small Dτ or at small or large elapsed times t. Associated with these fluctuations between particles, we also observe transient non-monotonicities of the brightness of individual particles. These non-monotonicities can be as much as 20% of the original brightness. We show that these novel photobleach-fluctuations dominate over variability of single-fluorophore orientation when there are at least 103 fluorophores on individual particles.

PEX16 contributes to peroxisome maintenance by constantly trafficking PEX3 via the ER.

The endoplasmic reticulum (ER) is required for the de novo biogenesis of peroxisomes in mammalian cells. However, its role in peroxisome maintenance is unclear. To explore ER involvement in the maintenance of peroxisomes, we redirect a peroxisomal membrane protein (PMP), PEX3, to directly target to the ER using the N-terminal ER signal sequence from preprolactin. Using biochemical techniques and fluorescent imaging, we find that ER-targeting PEX3 (ssPEX3) is continuously imported into pre-existing peroxisomes. This suggests that the ER constitutively provides membrane proteins and associated lipids to pre-existing peroxisomes. Using quantitative time-lapse live-cell fluorescence microscopy applied to cells that were either depleted of or exogenously expressing PEX16, we find that PEX16 mediates the peroxisomal trafficking of two distinct peroxisomal membrane proteins, PEX3 and PMP34, via the ER. These results not only provide insight into peroxisome maintenance and PMP trafficking in mammalian cells but also highlight important similarities and differences in the mechanisms of PMP import between the mammalian and yeast systems.

Uniform spatial distribution of collagen fibril radii within tendon implies local activation of pC-collagen at individual fibrils.

Collagen fibril cross-sectional radii show no systematic variation between the interior and the periphery of fibril bundles, indicating an effectively constant rate of collagen incorporation into fibrils throughout the bundle. Such spatially homogeneous incorporation constrains the extracellular diffusion of collagen precursors from sources at the bundle boundary to sinks at the growing fibrils. With a coarse-grained diffusion equation we determine stringent bounds, using parameters extracted from published experimental measurements of tendon development. From the lack of new fibril formation after birth, we further require that the concentration of diffusing precursors stays below the critical concentration for fibril nucleation. We find that the combination of the diffusive bound, which requires larger concentrations to ensure homogeneous fibril radii, and lack of nucleation, which requires lower concentrations, is only marginally consistent with fully processed collagen using conservative bounds. More realistic bounds may leave no consistent concentrations. Therefore, we propose that unprocessed pC-collagen diffuses from the bundle periphery followed by local C-proteinase activity and subsequent collagen incorporation at each fibril. We suggest that C-proteinase is localized within bundles, at fibril surfaces, during radial fibrillar growth. The much greater critical concentration of pC-collagen, as compared to fully processed collagen, then provides broad consistency between homogeneous fibril radii and the lack of fibril nucleation during fibril growth.

Single file diffusion into a semi-infinite tube.

We investigate single file diffusion (SFD) of large particles entering a semi-infinite tube, such as luminal diffusion of proteins into microtubules or flagella. While single-file effects have no impact on the evolution of particle density, we report significant single-file effects for individually tracked tracer particle motion. Both exact and approximate ordering statistics of particles entering semi-infinite tubes agree well with our stochastic simulations. Considering initially empty semi-infinite tubes, with particles entering at one end starting from an initial time t = 0, tracked particles are initially super-diffusive after entering the system, but asymptotically diffusive at later times. For finite time intervals, the ratio of the net displacement of individual single-file particles to the average displacement of untracked particles is reduced at early times and enhanced at later times. When each particle is numbered, from the first to enter (n = 1) to the most recent (n = N), we find good scaling collapse of this distance ratio for all n. Experimental techniques that track individual particles, or local groups of particles, such as photo-activation or photobleaching of fluorescently tagged proteins, should be able to observe these single-file effects. However, biological phenomena that depend on local concentration, such as flagellar extension or luminal enzymatic activity, should not exhibit single-file effects.

PEX5 and ubiquitin dynamics on mammalian peroxisome membranes.

Peroxisomes are membrane-bound organelles within eukaryotic cells that post-translationally import folded proteins into their matrix. Matrix protein import requires a shuttle receptor protein, usually PEX5, that cycles through docking with the peroxisomal membrane, ubiquitination, and export back into the cytosol followed by deubiquitination. Matrix proteins associate with PEX5 in the cytosol and are translocated into the peroxisome lumen during the PEX5 cycle. This cargo translocation step is not well understood, and its energetics remain controversial. We use stochastic computational models to explore different ways the AAA ATPase driven removal of PEX5 may couple with cargo translocation in peroxisomal importers of mammalian cells. The first model considered is uncoupled, in which translocation is spontaneous, and does not immediately depend on PEX5 removal. The second is directly coupled, in which cargo translocation only occurs when its PEX5 is removed from the peroxisomal membrane. The third, novel, model is cooperatively coupled and requires two PEX5 on a given importomer for cargo translocation--one PEX5 with associated cargo and one with ubiquitin. We measure both the PEX5 and the ubiquitin levels on the peroxisomes as we vary the matrix protein cargo addition rate into the cytosol. We find that both uncoupled and directly coupled translocation behave identically with respect to PEX5 and ubiquitin, and the peroxisomal ubiquitin signal increases as the matrix protein traffic increases. In contrast, cooperatively coupled translocation behaves dramatically differently, with a ubiquitin signal that decreases with increasing matrix protein traffic. Recent work has shown that ubiquitin on mammalian peroxisome membranes can lead to selective degradation by autophagy, or 'pexophagy.' Therefore, the high ubiquitin level for low matrix cargo traffic with cooperatively coupled protein translocation could be used as a disuse signal to mediate pexophagy. This mechanism may be one way that cells could regulate peroxisome numbers.

Cluster coarsening on drops exhibits strong and sudden size-selectivity.

Autophagy, an important process for degradation of cellular components, requires the targeting of autophagy receptor proteins to potential substrates. Receptor proteins have been observed to form clusters on membranes. To understand how receptor clusters might affect autophagy selectivity, we model cluster coarsening on a polydisperse collection of spherical drop-like substrates. Our model receptor corresponds to NBR1, which supports peroxisome autophagy. We recover dynamical scaling of cluster sizes, but find that changing the drop size distribution changes the cluster-size scaling distribution. The magnitude of this effect is similar to how changing the spatial-dimension affects scaling in bulk systems. We also observe a sudden onset of size-selection of the remaining drops with clusters, due to clusters evaporating from smaller drops and growing on larger drops. This coarsening-driven size selection provides a physical mechanism for autophagy selectivity, and may explain reports of size selection during peroxisome degradation.

A storage-based model of heterocyst commitment and patterning in cyanobacteria.

When deprived of fixed nitrogen (fN), certain filamentous cyanobacteria differentiate nitrogen-fixing heterocysts. There is a large and dynamic fraction of stored fN in cyanobacterial cells, but its role in directing heterocyst commitment has not been identified. We present an integrated computational model of fN transport, cellular growth, and heterocyst commitment for filamentous cyanobacteria. By including fN storage proportional to cell length, but without any explicit cell-cycle effect, we are able to recover a broad and late range of heterocyst commitment times and we observe a strong indirect cell-cycle effect. We propose that fN storage is an important component of heterocyst commitment and patterning in filamentous cyanobacteria. The model allows us to explore both initial and steady-state heterocyst patterns. The developmental model is hierarchical after initial commitment: our only source of stochasticity is observed growth rate variability. Explicit lateral inhibition allows us to examine ΔpatS, ΔhetN, and ΔpatN phenotypes. We find that ΔpatS leads to adjacent heterocysts of the same generation, while ΔhetN leads to adjacent heterocysts only of different generations. With a shortened inhibition range, heterocyst spacing distributions are similar to those in experimental ΔpatN systems. Step-down to non-zero external fN concentrations is also investigated.

Protein translocation without specific quality control in a computational model of the Tat system.

The twin-arginine translocation (Tat) system transports folded proteins of various sizes across both bacterial and plant thylakoid membranes. The membrane-associated TatA protein is an essential component of the Tat translocon, and a broad distribution of different sized TatA-clusters is observed in bacterial membranes. We assume that the size dynamics of TatA clusters are affected by substrate binding, unbinding, and translocation to associated TatBC clusters, where clusters with bound translocation substrates favour growth and those without associated substrates favour shrinkage. With a stochastic model of substrate binding and cluster dynamics, we numerically determine the TatA cluster size distribution. We include a proportion of targeted but non-translocatable (NT) substrates, with the simplifying hypothesis that the substrate translocatability does not directly affect cluster dynamical rate constants or substrate binding or unbinding rates. This amounts to a translocation model without specific quality control. Nevertheless, NT substrates will remain associated with TatA clusters until unbound and so will affect cluster sizes and translocation rates. We find that the number of larger TatA clusters depends on the NT fraction f. The translocation rate can be optimized by tuning the rate of spontaneous substrate unbinding, [Formula: see text]. We present an analytically solvable three-state model of substrate translocation without cluster size dynamics that follows our computed translocation rates, and that is consistent with in vitro Tat-translocation data in the presence of NT substrates.

An equilibrium double-twist model for the radial structure of collagen fibrils.

Mammalian tissues contain networks and ordered arrays of collagen fibrils originating from the periodic self-assembly of helical 300 nm long tropocollagen complexes. The fibril radius is typically between 25 to 250 nm, and tropocollagen at the surface appears to exhibit a characteristic twist-angle with respect to the fibril axis. Similar fibril radii and twist-angles at the surface are observed in vitro, suggesting that these features are controlled by a similar self-assembly process. In this work, we propose a physical mechanism of equilibrium radius control for collagen fibrils based on a radially varying double-twist alignment of tropocollagen within a collagen fibril. The free-energy of alignment is similar to that of liquid crystalline blue phases, and we employ an analytic Euler-Lagrange and numerical free energy minimization to determine the twist-angle between the molecular axis and the fibril axis along the radial direction. Competition between the different elastic energy components, together with a surface energy, determines the equilibrium radius and twist-angle at the fibril surface. A simplified model with a twist-angle that is linear with radius is a reasonable approximation in some parameter regimes, and explains a power-law dependence of radius and twist-angle at the surface as parameters are varied. Fibril radius and twist-angle at the surface corresponding to an equilibrium free-energy minimum are consistent with existing experimental measurements of collagen fibrils. Remarkably, in the experimental regime, all of our model parameters are important for controlling equilibrium structural parameters of collagen fibrils.

Dynamical network stability analysis of multiple biological ages provides a framework for understanding the aging process.

Widespread interest in non-destructive biomarkers of aging has led to a multitude of biological ages that each proffers a 'true' health-adjusted individual age. While each measure provides salient information on the aging process, they are each univariate, in contrast to the "hallmark" and "pillar" theories of aging which are explicitly multidimensional, multicausal and multiscale. Fortunately, multiple biological ages can be systematically combined into a multidimensional network representation. The interaction network between these biological ages permits analysis of the multidimensional effects of aging, as well as quantification of causal influences during both natural aging and, potentially, after anti-aging intervention. The behaviour of the system as a whole can then be explored using dynamical network stability analysis which identifies new, efficient biomarkers that quantify long term resilience scores on the timescale between measurements (years). We demonstrate this approach using a set of 8 biological ages from the longitudinal Swedish Adoption/Twin Study of Aging (SATSA). After extracting an interaction network between these biological ages, we observed that physiological age, a proxy for cardiometabolic health, serves as a central node in the network, implicating it as a key vulnerability for slow, age-related decline. We furthermore show that while the system as a whole is stable, there is a weakly stable direction along which recovery is slow -- on the timescale of a human lifespan. This slow direction provides an aging biomarker which correlates strongly with chronological age and predicts longitudinal decline in health - suggesting that it estimates an important driver of age-related changes.

Network dynamical stability analysis reveals key "mallostatic" natural variables that erode homeostasis and drive age-related decline of health.

Using longitudinal study data, we dynamically model how aging affects homeostasis in both mice and humans. We operationalize homeostasis as a multivariate mean-reverting stochastic process. We hypothesize that biomarkers have stable equilibrium values, but that deviations from equilibrium of each biomarker affects other biomarkers through an interaction network-this precludes univariate analysis. We therefore looked for age-related changes to homeostasis using dynamic network stability analysis, which transforms observed biomarker data into independent "natural" variables and determines their associated recovery rates. Most natural variables remained near equilibrium and were essentially constant in time. A small number of natural variables were unable to equilibrate due to a gradual drift with age in their homeostatic equilibrium, i.e. allostasis. This drift caused them to accumulate over the lifespan course and makes them natural aging variables. Their rate of accumulation was correlated with risk of adverse outcomes: death or dementia onset. We call this tendency for aging organisms to drift towards an equilibrium position of ever-worsening health "mallostasis". We demonstrate that the effects of mallostasis on observed biomarkers are spread out through the interaction network. This could provide a redundancy mechanism to preserve functioning until multi-system dysfunction emerges at advanced ages.

Network dynamical stability analysis reveals key "mallostatic" natural variables that erode homeostasis and drive age-related decline of health.

Using longitudinal study data, we dynamically model how aging affects homeostasis in both mice and humans. We operationalize homeostasis as a multivariate mean-reverting stochastic process. We hypothesize that biomarkers have stable equilibrium values, but that deviations from equilibrium of each biomarker affects other biomarkers through an interaction network - this precludes univariate analysis. We therefore looked for age-related changes to homeostasis using dynamic network stability analysis, which transforms observed biomarker data into independent "natural" variables and determines their associated recovery rates. Most natural variables remained near equilibrium and were essentially constant in time. A small number of natural variables were unable to equilibrate due to a gradual drift with age in their homeostatic equilibrium, i.e. allostasis. This drift caused them to accumulate over the lifespan course and makes them natural aging variables. Their rate of accumulation was correlated with risk of adverse outcomes: death or dementia onset. We call this tendency for aging organisms to drift towards an equilibrium position of ever-worsening health "mallostasis". We demonstrate that the effects of mallostasis on observed biomarkers are spread out through the interaction network. This could provide a redundancy mechanism to preserve functioning until multi-system dysfunction emerges at advanced ages.

Modelling lifespan reduction in an exogenous damage model of generic disease.

We model the effects of disease and other exogenous damage during human aging. Even when the exogenous damage is repaired at the end of acute disease, propagated secondary damage remains. We consider both short-term mortality effects due to (acute) exogenous damage and long-term mortality effects due to propagated damage within the context of a generic network model (GNM) of individual aging that simulates a U.S. population. Across a wide range of disease durations and severities we find that while excess short-term mortality is highest for the oldest individuals, the long-term years of life lost are highest for the youngest individuals. These appear to be universal effects of human disease. We support this conclusion with a phenomenological model coupling damage and mortality. Our results are consistent with previous lifetime mortality studies of atom bomb survivors and post-recovery health studies of COVID-19. We suggest that short-term health impact studies could complement lifetime mortality studies to better characterize the lifetime impacts of disease on both individuals and populations.